We conclude this note with the following discussion about complex projective (or compact Kähler) manifolds with nef tangent bundles. Let \(X\)  be a complex projective manifold (or even a Kähler manifold) whose tangent bundle is nef. Examples include rational homogeneous manifolds (e.g., complex projective spaces, flag manifolds, or quotients \(G/P\)  of a simply connected complex Lie group \(G\)  by a parabolic subgroup), abelian varieties (or complex tori), etc. (cf. [1]}). Moreover, as we will argue below, complex projective (or compact Kähler) manifolds with non-negative sectional curvature have nef tangent bundles.
By applying Theorem REF  to the tangent bundle \(TX\) , one gets via the Gauss-Bonnet formula that
\( \chi (X)=\int _X c_n(TX)=T_XX \cdot T_XX \ge 0,\) 
